Lecture 10: The Internal Structure of Stars

Key Ideas:

• Observational Clues to Stellar Structure:
  • H-R Diagram
  • Mass-Luminosity Relationship
• Hydrostatic Equilibrium
  • Balance between Gravity & Pressure
• Core-Envelope Structure of Stars
  • Hot, dense, compact core
  • cooler, low-density, extended envelope

From Stellar Properties to Stellar Structure

Any theory of stellar structure must explain the observed properties of stars.

Seek clues in correlations among the observed properties, in particular:

• Mass
• Luminosity
• Radius
• Temperature

The H-R Diagram

Main Sequence:

• Strong correlation between Luminosity and Temperature.
• Holds for 85% of nearby stars including the sun

All other stars differ in size:

Giants & Supergiants:

• Very large radius, but same masses as M-S stars

White Dwarfs:

• very compact (~R_earth), but only ~M_sun

Mass-Luminosity Relationship

For Main-Sequence stars: L~M^3.5

In words:

"More massive M-S stars are more luminous."

This is not true for Giants or White Dwarfs.

Stellar Density

Density = Mass/Volume

• Sun: ~1.6 g/cc
• O5v Star: ~0.005 g/cc
• M0v Star: ~5 g/cc

Giants: 10^-7 g/cc

White Dwarfs: 10⁵ g/cc

Interpreting the Observations:

For Main-Sequence Stars:

• Strong L-T Relationship on H-R Diagram
• Strong M-L Relationship

Means they have similar structures & governing laws.

Giants and White Dwarfs must have very different internal structures from M-S stars of similar mass.

Laws of Stellar Structure I: The Gas Law

Most stars obey the Perfect Gas Law:

Pressure = density x Temperature

In words:

• Compressing a gas results in higher P & T
• Expanding a gas results in lower P & T

Laws of Stellar Structure II: The Law of Gravity

Stars are very massive & bound together by their Self-Gravity

Gravitational binding increases as 1/R²

In words:

• Compress a star, gravitational binding stronger.
• Expand a star, gravitational binding gets weaker.

Hydrostatic Equilibrium

Gravity & Pressure play opposite roles:

• Gravity wants to make a star contract.
• Pressure wants to make a star expand.

Counteract each other:

• Gravity confines the gas against Pressure.
• Pressure supports the star against Gravity.

Exact Balance = Hydrostatic Equilibrium

The star neither expands nor contracts.

Core-Envelope Structure

The deeper you go into a star, the greater the pressure.

The Gas Law says:

"More pressure= hotter, denser gas"

Consequences:

• hot, dense, compact CORE
• cooler, lower density, extended ENVELOPE

Example: The Sun

• Radius = 0.25 R_sun
• T = 15 Million K
• Density = 150 g/cc

• Radius = R_sun = 700,000 km
• T = 5800 K
• Density = 10^-7 g/cc

The Essential Tension

The Life of a star is a constant tug-of-war between Gravity & Pressure.

Tip the internal balance either way, and it will change the star's outward appearance:

Internal Changes have External Consequences